完全三部图K_（2，2，2）的弧传递Z_n-正则覆盖
摘要
设Г是有限无向简单正则图.若Г没有孤立点,我们称图Г是弧传递的或对称的,如果Г的自同构群Aut(Г)传递地作用在Г的弧集合上.本文讨论了完全三部图K_(2,2,2)的弧传递Z_n-正则覆盖,得到了几类新的4度对称图,其中n=4k或n=4k且k≡2(mod 4).特别地,我们得到了一类新的点稳定子无界并且不是两图的字典式积的4度对称图.
LetГbe a finite undirected simple regular graph which has no isolated vertices.We say thatГis arc-transitive or symmetric graph,if its full automorphism group Aut(Г) acts transitively on its arc set.In this paper,we investigate the arc-transitive Z_n-regular coverings of K_(2,2,2),and obtain several new infinite families of 4-valent symmetric graphs as the covering graphs of K_(2,2,2) by Z_n,where n=4k or n=4k and k=2(mod 4).In particular,we obtain an infinite family of 4-valent symmetric graphs,which are not a lexicographic product of two graphs and has unboundary order of the vertex-stabilizer.
LetГbe a finite undirected simple regular graph which has no isolated vertices.We say thatГis arc-transitive or symmetric graph,if its full automorphism group Aut(Г) acts transitively on its arc set.In this paper,we investigate the arc-transitive Z_n-regular coverings of K_(2,2,2),and obtain several new infinite families of 4-valent symmetric graphs as the covering graphs of K_(2,2,2) by Z_n,where n=4k or n=4k and k=2(mod 4).In particular,we obtain an infinite family of 4-valent symmetric graphs,which are not a lexicographic product of two graphs and has unboundary order of the vertex-stabilizer.
引文
[1]Dixon J.D.and Mortimer B.,Permutation Groups,Springer-Verlag,New York,1996
[2]Wielandt,H.,Finite Permutation Groups,Academic Press,New York,1964
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[10]Y.Cheng and J.Oxley,On weakly symmetric graphs of order twice a prime,J.Comb.Theory B,42(1987),196-211
[11]R.J.Wang and M.Y.Xu,A classification of symmetric graphs of order 3p,J.Combin.Theory Ser.B,58(1993),197-216
[12]C.E.Praeger,R.J.Wang and M.Y.Xu,Symmetric graphs of order a product of two distinct primes,J.Combin.TheorySer.B,58(1993),299-318
[13]Gross J.L.and Tucker T.W.,Generating all graph covering by permutation voltage assignment,Discrete Math.,18(1977),273-283
[14]Feng Y.Q.and Kwak.J.H.,Constructing an infinite family of 1-regular graphs,Europ.J.Combin.,23(2002),559-565
[15]Feng Y.Q.and Kwak.J.H.,S-regular cycle coverings of the complete bipartite graph K_(3,3),J.Graph Theory(2002) in press.
[16]Feng Y.Q.and Wang K.S.,S-regular cycle coverings of symmetric the threedimensional hypercube Q_3,Europ.J.Combin.,23(2002),1-13
[17]Malnic A.,Group actions,coverings and lifts of automorphisms,Discrete Math.,182(1998),203-218
[18]Wang C.Q.and Chen T.S.,Semisymmetric cubic graphs as regular covers of K_(3,3),Acta Mathematic Sinica,24(3)(2008),405-416
[19]Wang C.Q.and Xu M.Y.,Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D_(2n),Europ.J.Combin.,27(2006),750-766
[20]Djokovic D.Z.,Automorphisms of graphs and coverings,J.Combin Theory B,16(1974),243-247
[21]Malnic A.,Nedela R.and Skoviera M.,Lifting graph automorphisms by voltage assignments,Europ.J.Combin.,21(2000),927-947
[22]M.Conder,A.Malnic,D.Marusic and P.Potocnik,A census of semisymmetric cubic graphs on up to 768 vertices,J.Algebr.Comb.,23(2006),255-294
[23]M.Skoviera,A construction to the theory of voltage groups,Discrete Math.,61(1986),281-292
[24]Feng Y.Q.and Wang K.S.,S-regular cycle coverings of the three-dimensional hypercube Q_3,Europ.J.Combin.,24(2003),719-731
[2]Wielandt,H.,Finite Permutation Groups,Academic Press,New York,1964
[3]Gorenstein D.,Finite Groups,Harper and Row,New York,1968
[4]Folkman J.,Regular line-symmetric graghs,J.Combin.Theory,3(1967),215-232
[5]Du S.F.and Xu M.Y.,A classification of semisymmetric gragh of order 2pq,Com.in Algbra,28(6)(2000),2685-2715
[6]Z.P.Lu,C.Q.Wang and M.Y.Xu,On semisymmetric cubic graphs of order 6p~2,Sciencein China Ser.A,(1)(2004),11-17
[7]Malinc A.,Marusic D.and Wang C.Q.,Cubic edge-transitive graphs of 2p~3.Discrete Mathematics,274(2004),187-198
[8]Wang C.Q.,Semisymmetric cubic graph of order 2p~2q[C],Com.Mac Preprint Series.,(2002)
[9]C.Y.Chao,On the classification of symmetric graphs with a prinic number of vertices,Trans.Amer.Math.Soc.,158(1971),247-256
[10]Y.Cheng and J.Oxley,On weakly symmetric graphs of order twice a prime,J.Comb.Theory B,42(1987),196-211
[11]R.J.Wang and M.Y.Xu,A classification of symmetric graphs of order 3p,J.Combin.Theory Ser.B,58(1993),197-216
[12]C.E.Praeger,R.J.Wang and M.Y.Xu,Symmetric graphs of order a product of two distinct primes,J.Combin.TheorySer.B,58(1993),299-318
[13]Gross J.L.and Tucker T.W.,Generating all graph covering by permutation voltage assignment,Discrete Math.,18(1977),273-283
[14]Feng Y.Q.and Kwak.J.H.,Constructing an infinite family of 1-regular graphs,Europ.J.Combin.,23(2002),559-565
[15]Feng Y.Q.and Kwak.J.H.,S-regular cycle coverings of the complete bipartite graph K_(3,3),J.Graph Theory(2002) in press.
[16]Feng Y.Q.and Wang K.S.,S-regular cycle coverings of symmetric the threedimensional hypercube Q_3,Europ.J.Combin.,23(2002),1-13
[17]Malnic A.,Group actions,coverings and lifts of automorphisms,Discrete Math.,182(1998),203-218
[18]Wang C.Q.and Chen T.S.,Semisymmetric cubic graphs as regular covers of K_(3,3),Acta Mathematic Sinica,24(3)(2008),405-416
[19]Wang C.Q.and Xu M.Y.,Non-normal one-regular and 4-valent Cayley graphs of dihedral groups D_(2n),Europ.J.Combin.,27(2006),750-766
[20]Djokovic D.Z.,Automorphisms of graphs and coverings,J.Combin Theory B,16(1974),243-247
[21]Malnic A.,Nedela R.and Skoviera M.,Lifting graph automorphisms by voltage assignments,Europ.J.Combin.,21(2000),927-947
[22]M.Conder,A.Malnic,D.Marusic and P.Potocnik,A census of semisymmetric cubic graphs on up to 768 vertices,J.Algebr.Comb.,23(2006),255-294
[23]M.Skoviera,A construction to the theory of voltage groups,Discrete Math.,61(1986),281-292
[24]Feng Y.Q.and Wang K.S.,S-regular cycle coverings of the three-dimensional hypercube Q_3,Europ.J.Combin.,24(2003),719-731